adopt, or apply the principle of cancelling to the ordinary mode of state. ment. It will be well to employ both modes, as together they open a wider field for illustration. It is sometimes remarked of the cancelling system, that it is good as far as it goes. The same may be said of arithmetic; for the principle is inseparable from it. It is the only principle by which any question in division can be performed. Wherever it cannot be applied, the numbers must be written in the form of a fraction. When the question involves multiplication and division, it will generally be found to be a great saving of labor, to write down all those numbers which are to be factors of the dividend and divisor, before proceeding to the operation. The eye will then detect at a glance equal factors, and they can be excluded from the operation. The teacher will bear in mind the importance of giving general illustrations of arithmetical principles, whenever it can be done, as its tendency is to enlarge the views of the pupil and to give importance to the study. For example, let simple division be illustrated not only arithmetically, but on general principles. Let it be required to divide 16 by 8, and it may be done and illustrated in the following manner:— 8)16-8X2-8=2 Ans. Now substitute the letter a for 8, and the letter b for 2, and read the question thus: divide ab by a. sor. b Ans. Here, as before, we exclude from the dividend a factor equal to the diviBut this latter process is algebraic; hence the scholar's views are extended, and he perceives at once, and for the first time, the connection between arithmetic and algebra. Formulas are also given to aid the less experienced teacher, and also to bring out more prominently arithmetical principles. MANNER OF RECITATION. Promptness and dispatch are characteristics of our times, and young men must be educated in reference to them. There is no place, perhaps, better calculated to train a scholar to think and act with precision and energy, than at the blackboard. When a scholar is called out from his class to solve a question, let him quickly, and with gentlemanly mien endeavoring to be self-possessed, take his stand at the board, read his question distinctly, and with the same reference to rhetorical notation as though he were called out on purpose for the reading of the question. Then let him state his question, giving the reasons for each step as he proceeds; or let him state and solve his question, then return to the commencement, and illustrate the principle, and give the reason for each step in the solution. Then let him pause at the board a moment, for his teacher to propose such questions as he may think proper. A brief view has now been given of the plan and mode of teaching arithmetic adopted in this system. It is confidently believed, from the long experience the author has had in teaching, that the mode here adopted for presenting the subject of arithmetic, will be found better calculated to induce a fondness for the study; that it unfolds more of the science, and brings out principles more clearly than any other system now before the public. With these views the author submits the work to the candid perusal of all who are interested in the progress of knowledge. CHARLES G. BURNHAM. DANVILLE, VT., Oct. 18, 1849. DEFINITIONS, AXIOMS, AND SIGNS. DEFINITIONS. A definition is what is meant by a word or phrase. The language of a definition should be so plain as not to be capable of misapprehension. 1. Quantity is any thing which may be multiplied, divided, and measured. 2. Magnitude is that species of quantity which is extended; i. e. which has one or more of the three dimensions-length, breadth, and thickness. A line is a magnitude, because it has length. 3. Mathematics is the science of quantity. 4. Arithmetic is the science of numbers. 5. Algebra is a method of computing by letters and other symbols. 6. Geometry treats of lines, surfaces, and solids. Arithmetic, Algebra, and Geometry are those parts of mathematics, on which all the others are founded. 7. A Demonstration is a course of reasoning which establishes a truth. 8. A Proposition is any thing proposed: if to be proved or demonstrated, it is called a Theorem; if to be done, it is called a Problem. 9. A plus quantity is a quantity to be added, and has this sign + before it; thus, +6. 10. A minus quantity is a quantity to be subtracted, and has this sign - before it; thus, 6. 11. An Equation is a proposition expressing equality between one quantity, or set of quantities, and another, or between different expressions for the same quantity; thus, 5=3+2. 12. A member of an equation is the quantity or quantities on one side of the sign of equality. OBS.-For definitions of terms in more common use in this work, see Art. 54, or Part I. AXIOMS An axiom is a self-evident proposition. 1. Things which are equal to the same thing are equal to each other. 2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal. 4. If equals be added to unequals, the wholes will be unequal. 5. If equals be taken from unequals, the remainders will be unequal 6. Things which are double of equal things are equal to each other. 7. Things which are halves of the same thing, are equal to each other 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts. SIGNS. Equality is denoted by two horizontal lines. +Addition: as 4+3=7; which signifies that 4 added to 3 equals 7. × Multiplication: as 4X3=12; which signifies that 4 multiplied by 3 equals 12. Subtraction: as 4-3=1; which signifies that 3 taken from 4 leaves 1. 2 (,,, 214, Division: as, 2)4(2, and 4÷2=2, and 1=2, and 2|4=2. In either case it signifies that 4 divided by 2 equals 2. :::: Proportion: as, 2: 4 :: 6:12; which is read, 2 is to 4 as 6 is to 12. Vinculum: as 4+3=7; which is read, the sum of 4 and 3 equals 7, and 4—3—1, is read, the difference of 4 and 3 equals 1. Radical sign: placed before a number denotes that the square root is to be taken. 42 implies that 4 is to be raised to the second power. 43 implies that 4 is to be raised to the third power. implies the third root. Reduction of Fractions .............. 118 Reduction Descending and Ascending 121 |